2.2 Monte Carlo Simulations
2.2.2 Simulation Results
In Table 2.1, I report the average absolute value of bias and RMSE over 10 ex-periments for three different designs: T = {15, 30, 60}. The results of exex-periments in which the correlation between the time-varying variable x3 and unit effects δ is varied are displayed from the second column to the seventh column; those in which the correlation between the rarely changing variable w3 and unit effects δ is varied are displayed in the eighth column to the thirteenth column. Notice that in the former corr(w3, δj) is fixed at the value of 0.3 while in the latter corr(x3, δj) is fixed at the value of 0.3.
These results in Table 2.1 are summarized as follows. First, on average, the pooled OLS produces the poorest estimates in terms of the absolute bias and RMSE.
As we know and the results confirm, the correlation between the LDV and unit effects would seriously bias the pooled OLS estimator. Furthermore, the correlation between covariates and unit effects exacerbates the problem. The magnitude of bias and RMSE does not decrease as T increases.
Second, among these six estimators, the FEVD estimator produces estimates with the largest bias and RMSE for the rarely changing variable in all cases while its estimates for the LDV and the time-varying variable are exactly the same with those under the FE model. The results indicate that the FEVD does not perform well for estimating rarely changing variables in dynamic panel data models as Pl¨umper and Troeger (2007) claim.
Table 2.1: Average Absolute Bias and RMSE
corr(x3, δ) = {0, · · · , 0.9} corr(w3, δ) = {0, · · · , 0.9}
Bias RMSE Bias RMSE
yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3
OLS 0.015 0.228 0.095 0.017 0.241 0.119 0.016 0.131 0.160 0.018 0.149 0.185
FE 0.006 0.063 0.126 0.007 0.081 0.159 0.006 0.052 0.129 0.007 0.065 0.166
FEVD 0.006 0.063 0.353 0.007 0.081 0.393 0.006 0.052 0.387 0.007 0.065 0.421
RE 0.004 0.099 0.103 0.005 0.113 0.126 0.005 0.050 0.159 0.006 0.063 0.183
MLM 0.005 0.063 0.121 0.006 0.081 0.153 0.005 0.052 0.124 0.006 0.065 0.160
BSEM 0.005 0.063 0.113 0.006 0.080 0.142 0.005 0.052 0.119 0.006 0.065 0.151
corr(w3, δ) = 0.3, T = 15 corr(x3, δ) = 0.3, T = 15
yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3
OLS 0.018 0.251 0.112 0.019 0.261 0.140 0.019 0.146 0.155 0.021 0.159 0.180
FE 0.004 0.044 0.083 0.005 0.054 0.104 0.004 0.034 0.080 0.005 0.042 0.100
FEVD 0.004 0.044 0.311 0.005 0.054 0.345 0.004 0.034 0.351 0.005 0.042 0.380
RE 0.003 0.070 0.079 0.004 0.080 0.098 0.004 0.033 0.108 0.005 0.041 0.127
MLM 0.003 0.044 0.081 0.004 0.054 0.103 0.004 0.034 0.078 0.004 0.042 0.098
BSEM 0.003 0.043 0.080 0.004 0.054 0.100 0.004 0.034 0.077 0.004 0.042 0.096
corr(w3, δ) = 0.3, T = 30 corr(x3, δ) = 0.3, T = 30
yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3 yt−1 x3 w3
OLS 0.021 0.265 0.156 0.023 0.276 0.189 0.023 0.159 0.185 0.024 0.171 0.220
FE 0.004 0.030 0.065 0.005 0.037 0.083 0.003 0.023 0.064 0.004 0.029 0.080
FEVD 0.004 0.030 0.338 0.005 0.037 0.366 0.003 0.023 0.351 0.004 0.029 0.376
RE 0.003 0.044 0.065 0.004 0.052 0.083 0.003 0.023 0.081 0.004 0.029 0.098
MLM 0.003 0.030 0.064 0.004 0.037 0.081 0.003 0.023 0.063 0.004 0.029 0.079
BSEM 0.003 0.030 0.063 0.004 0.037 0.081 0.003 0.023 0.063 0.004 0.029 0.078
corr(w3, δ) = 0.3, T = 60 corr(x3, δ) = 0.3, T = 60
Third, the RE is more biased than the FE in estimating correlated covariates x3 and w3 while the RE is more efficient than the FE in estimating slightly correlated rarely changing variable w3. However, as the correlation is modeled under the MLM, the MLM and the FE perform equally well in estimating coefficients for the LDV and
correlated covariates in terms of bias and RMSE. In other words, when the correlation between unit effects and covariates is modeled under the random-effects framework in the way suggested by Mundlak (1978), the MLM is less biased at the expense of efficiency.
Last but not least, the BSEM performs as well, or better than the FE and MLM in terms of bias and RMSE. We can see that the BSEM is as less biased as the FE in all cases and is more efficient than the FE and MLM when T is less than 30. In other words, the proposed model, which explicitly accounts for the correlation between unit effects and covariates, is less biased than the RE and more efficient than the FE.
These properties of the proposed model are remarkable especially when the sample size is small.
I then show the effects of the correlation on the absolute bias and RMSE of the estimates for the LDV, time-varying variable x3, and rarely changing variable w3. To save space, I only present the results from the design where T = 15, but the results remain the same when T = {30, 60}. Figure 2.2 presents the absolute value of bias and the RMSE of the six estimators for the LDV, x3and w3when corr(x3, δj) is varied and corr(w3, δj) is fixed at the value of 0.3. The three panels on the left-hand side ((a), (c), and (e)) show the absolute values of bias and, as can be seen, the pooled OLS produces estimates of the LDV and x3 with the largest bias. The RE does not perform well for correlated covariate x3 and gets worse as the size of the correlation between unit effects and the time-varying variable increases. The FEVD estimator has the poorest estimates of rarely changing variables and the FE, MLM, and BSEM
(a) Absolute Bias of LDV
(c) Absolute Bias of x3
Absolute Bias
(e) Absolute Bias of w3
Absolute Bias
perform more or less equally well in estimating all of these three variables no matter what the size of the correlation is.
As can be seen, the same pattern appears in the RMSE of these estimators resented in the three panels on the right-hand side of Figure 2.2. Simply put, the pooled OLS has the largest RMSE for the LDV and time-varying variable x3 while the FEVD estimator performs poorly for the estimate of rarely changing variable w3. The BSEM performs a little bit better than the FE and MLM in terms of RMSE. These results hold no matter what the size of the correlation between unit effects and the rarely changing variable is.
(a) Absolute Bias of LDV
(c) Absolute Bias of x3
Absolute Bias FE, FEVD, RE, MLM, BSEM
(d) RMSE of x3 FE, FEVD, RE, MLM, BSEM
(e) Absolute Bias of w3
Absolute Bias
Figure 2.3 presents the absolute value of bias and the RMSE of the six estima-tors for the LDV, time-varying variable x3 and rarely changing variable w3 when corr(x3, δj) is fixed at the value of 0.3 and corr(w3, δj) is varied. Figure 2.3 shows that, in general, the pooled OLS has largest biased and inefficient estimates of the LDV and x3; the FEVD perform poorly in terms of the bias and RMSE in estimating the coefficient of w3. The FE, MLM, and BSEM perform equally well in estimating the coefficients of the LDV and the correlated time-varying variable x3 and rarely changing variable w3.
Finally, I present the estimates of correlation between unit effects and the time-varying variable and rarely changing variable from the proposed model. To save the
+
Figure 2.4.: The 80% Highest Posterior Density Intervals of the Estimates of the Correla-tion between Unit Effects and Covariates.
space, I only show the results from the design where T = 15 in Figure 2.4. The two panels on the top are from the design where corr(x3, δj) is varied while corr(w3, δj) is fixed at the value of 0.3; the two panels at the bottom are from the design where corr(w3, δj) is varied while corr(x3, δj) is fixed at the value of 0.3. In this figure, black lines represent the 80% highest posterior density (HPD) intervals of the correlation between unit effects and covariates from generated data while blue lines represent the
80% HPD intervals of the correlation estimates. As can be seen in Figure 2.4, the estimates of correlation between unit effects and the time-varying variable and rarely changing variable are approximately close to those in the generated data.
In short, the proposed model not only perform as well, or better than classical estimators in estimating coefficients for correlated covariates in terms of bias and effi-ciency but also provide approximately accurate estimates for correlation between unit effects and covariates. Therefore, the proposed model is preferred when the sample size is small and the degree of the correlation between unit effects and covariates is of interest rather than eliminating unit effects such as the FE or assuming no correlation like the RE.